Binomial Theorem
Expand (a+b)ⁿ
Expand (a + b)ⁿ
About This Calculator
The binomial theorem provides a formula for expanding powers of a binomial (a+b)ⁿ without repeated multiplication. The coefficients of each term are exactly the entries of Pascal's triangle (binomial coefficients). This theorem is used in algebra, probability (binomial distribution), and calculus.
Formula
(a+b)ⁿ = Σ C(n,k) × aⁿ⁻ᵏ × bᵏ for k = 0 to n
C(n,k) = n! / (k!(n−k)!) is the binomial coefficient
(a+b)² = a²+2ab+b²; (a+b)³ = a³+3a²b+3ab²+b³
Example Calculation
Expand (x+2)⁴.
- Use C(4,0)x⁴ + C(4,1)x³(2) + C(4,2)x²(4) + C(4,3)x(8) + C(4,4)(16)
- = 1·x⁴ + 4·2x³ + 6·4x² + 4·8x + 1·16
- = x⁴ + 8x³ + 24x² + 32x + 16
(x+2)⁴ = x⁴ + 8x³ + 24x² + 32x + 16
Binomial Expansions
| Power | Expansion | Coefficients |
|---|---|---|
| (a+b)¹ | a + b | 1, 1 |
| (a+b)² | a²+2ab+b² | 1, 2, 1 |
| (a+b)³ | a³+3a²b+3ab²+b³ | 1, 3, 3, 1 |
| (a+b)⁴ | a⁴+4a³b+6a²b²+4ab³+b⁴ | 1, 4, 6, 4, 1 |
| (a+b)⁵ | ... | 1, 5, 10, 10, 5, 1 |
Frequently Asked Questions
What is the binomial distribution?
The binomial distribution gives the probability of exactly k successes in n independent trials with probability p of success each time. P(X=k) = C(n,k) × pᵏ × (1−p)^(n−k). For example, flipping a fair coin 10 times.
What is the general term in a binomial expansion?
The (k+1)-th term of (a+b)ⁿ is C(n,k) × aⁿ⁻ᵏ × bᵏ. This lets you find any specific term without expanding the entire expression.
Can the binomial theorem extend to fractional or negative exponents?
Yes, via the generalized binomial series: (1+x)ⁿ = 1 + nx + n(n-1)x²/2! + ... This infinite series converges for |x| < 1 and is used in approximations like √(1+x) ≈ 1 + x/2 for small x.
How do I quickly find the coefficient of a specific term?
For the term containing bᵏ in (a+b)ⁿ, the coefficient is C(n,k). For example, the coefficient of x²y³ in (x+y)⁵ is C(5,3) = 10.